For any two graphs \(G\)  and \(H\) , the Ramsey number \(r(G,H)\)  is the smallest positive integer \(n\)  such that \(K_n\rightarrow (G,H)\) . The classical Ramsey number \(r(s, t)\)  is defined as \(r(K_s,K_t)\)  and the diagonal Ramsey number is defined as \(r(K_n,K_n)\) . These numbers have been studied extensively in the last five decades. The difficulty in exact determination of diagonal Ramsey number \(r(n, n)\) , swifts expeditiously from the apparent \(r(3,3)=6\)  to the unmanageable \(r(5,5)\) . Currently, the best known lower and upper bounds for \(r(5,5)\)  are 43 and 48 (see [1]}). Many interesting variations of the basic problem of finding classical Ramsey numbers have emerged. One such variation is the calculation of the number of Ramsey critical \((G,H)\)  graphs, for any pair of graphs \((G,H)\) . In this paper, we show that there are exactly 68 Ramsey critical \(r(C_n, K_6)\)  graphs, for all \(n\)  exceeding fourteen.
